In variational autoencoders, what does p(x|z) mean?
Whilst you're right that for any continuous distribution $P(X = x) = 0 \;; \forall x \in \mathcal{X}$ where $\mathcal{X}$ is there support of the distribution, they are not referring to probabilities here, rather they are referring to density functions (though this should really be denoted with a lower case $p$ to avoid confusion such as this).
$p(x|z)$ is a conditional distribution, which is also allowed in the continuous case -- you can also 'mix and match', i.e. $x$ could be continuous and $z$ could be discrete, and vice-versa.
In the paper, all the authors are meaning when they write $p(x|z)$ is the density of $x$ conditioned on $z$; in VAE's with an image application this is the conditional density of the image $x$ given your latent vector $z$.